Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $a \neq 0$. $y = \dfrac{2}{40a - 16} \div \dfrac{3a}{6a(5a - 2)} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $y = \dfrac{2}{40a - 16} \times \dfrac{6a(5a - 2)}{3a} $ When multiplying fractions, we multiply the numerators and the denominators. $y = \dfrac{ 2 \times 6a(5a - 2) } { (40a - 16) \times 3a } $ $ y = \dfrac {2 \times 6a(5a - 2)} {3a \times 8(5a - 2)} $ $ y = \dfrac{12a(5a - 2)}{24a(5a - 2)} $ We can cancel the $5a - 2$ so long as $5a - 2 \neq 0$ Therefore $a \neq \dfrac{2}{5}$ $y = \dfrac{12a \cancel{(5a - 2})}{24a \cancel{(5a - 2)}} = \dfrac{12a}{24a} = \dfrac{1}{2} $